Section 12.3

Equations of Planes

A plane is defined by a single point and a perpendicular "normal" vector.

1

Introduction

To define a line, we needed a parallel vector. To define a plane, we need a Normal Vector that is perpendicular to the surface. The equation works because the dot product of with any vector lying in the plane must be zero.

Interactive: Plane and Normal Vector

The red normal vector is perpendicular to every direction in the blue plane.

2

Key Formulas

Scalar Equation of a Plane

Through with normal :

Linear Equation

Standard form:

(Here, is directly visible!)

3

Worked Examples

Example 1: Finding the Equation (Level 1)

Find the plane passing through with normal vector .

Step 1: Substitute into Scalar Form

Step 2: Expand and Simplify



Example 2: Plane through Three Points (Level 2)

Find the plane containing points , , and .

Strategy: We need a normal vector . We can find one by taking the cross product of two vectors in the plane, like and .

1. Find Vectors

2. Cross Product (Normal Vector)

From Lesson 11.4:


3. Write Equation

Use point P(1,0,0) and normal :

Example 3: Line Orthogonal to Plane? (Level 3)

Is the line orthogonal to the plane ?

  • Get Direction Vectors:
    Line direction .
    Plane normal .
  • Condition:
    The line is orthogonal to the plane if the line's vector is parallel to the plane's normal .
  • Check Parallelism:
    Is a scalar multiple of ?
    .
    Yes! Since , they are parallel.
  • Conclusion: The line is orthogonal to the plane.
4

Practice Quiz

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